Question

Do bonds reduce the overall risk of an investment portfolio? Let x be a random variable representing annual percent return for Vanguard Total Stock Index (all stocks). Let y be a random variable representing annual return for Vanguard Balanced Index (60% stock and 40% bond). For the past several years, we have the following data.

x: 35 0 11 11 12 22 13 −23 −9 −13 y: 23 −2 13 15 18 22 12 −9 −6 −3

(a) Compute Σx, Σx2, Σy, Σy2.

(b) Use the results of part (a) to compute the sample mean, variance, and standard deviation for x and for y. (Round your answers to two decimal places.)

(c) Compute a 75% Chebyshev interval around the mean for x values and also for y values. (Round your answers to two decimal places.)

(d) Compute the coefficient of variation for each fund. (Round your answers to the nearest whole number.)

Answer 1

From the given problem X is random variable represents the Annual percentage of return for Vanguard total stock index

And Y is the random variable representing the annual return vanguard balanced index

a) from the given data X = 59   X² = 3043

Y = 83 Y² = 2005

b) Mean of X =  X / n = 59/10 = 5.9

Mean of Y =  Y /n = 83/10 = 8.3

variance of X =  (X²/ n ) - (X / n )²

- = (3043/10) - ( 5.9)² = 269.49

   Standard deviation of X = 16.42

Vriance of Y = (Y²/ n ) -   (Y / n )²

   ^{= (2005/10) - (8.3)2= 131.61}

^{Standard deviation of Y = 11.47}

C. Chebychevs interval around mean

   we observe that 1- 1/k² = 0.75

     1/k² = 0.25

k = 2

1)75% chebychevs interval around mean for X is given as (Mean- 2 S.D , Mean + 2S.D)

(- 26.94 , 38.74 )

2) 75% chebychevs interval around the mean of Y is given as (Mean- 2 S.D , Mean + 2S.D)

(- 14.64 , 31.24 )

(d) Coefficient of variation for X = (standard deviation of X / Mean of X ) % = (16.42 / 5.9) % = 2.78% = 3% ( approximately)

Coefficient of variation for Y = (Standard deviation of Y / Mean of Y) %

= (11.47/ 8.3 ) % = 1.38 % =1 % ( approximately)

Coefficient of variation for y is less than coefficient of variation for X

Hence we conclude that fund Y maintains more consistency than Fund X